# Rational equivalence of algebras, its clone generalizations, and clone categoricity

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## Abstract

The clones of functions on a set *A* are related to an arbitrary universal algebra \(\mathfrak{A} = \left\langle {A;\sigma } \right\rangle \) in various natural ways. The simplest and minimal of them is the clone \(Tr(\mathfrak{A})\) of termal functions of the algebra \(\mathfrak{A}\), that is, the closure of the collection of signature functions of these algebras with respect to the operator of superposition. The coincidence of similar clones (up to conjugation by the bijections of the underlying sets of the algebras) generates various equivalence relations on universal algebras, the first of which is the relation of rational equivalence introduced by Mal’tsev. This article deals with clone equivalences of this kind between algebras of arbitrary signatures.

## Keywords

clone of functions rational equivalence of algebras derived structures of algebras clone categoricity of algebras## Preview

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## References

- 1.
*Mal’tsev A. I.*, “Structure characterization of some classes of algebras,” Dokl. Akad. Nauk SSSR,**120**, No. 1, 29–32 (1958).MathSciNetzbMATHGoogle Scholar - 2.
*Pinus A. G.*, “On conditional terms and identities on universal algebras,” Vychisl. Sistemy,**156**, 59–78 (1996).MathSciNetzbMATHGoogle Scholar - 3.
*Pinus A. G.*, “Inner homomorphisms and positively conditional terms,” Algebra and Logic,**40**, No. 2, 87–95 (2001).MathSciNetCrossRefGoogle Scholar - 4.
*Pinus A. G.*, Conditional Terms and Their Applications to Algebra and the Theory of Computations [in Russian], Izdat. NGTU, Novosibirsk (2002).Google Scholar - 5.
*Pinus A. G.*, “Conditional terms and their applications in algebra and computation theory,” Russian Math. Surveys,**56**, No. 4, 649–686 (2001).MathSciNetzbMATHCrossRefGoogle Scholar - 6.
*Pinus A. G.*, “Conditional identity calculus and the conditioned rational equivalence,” Algebra and Logic,**37**, No. 4, 245–259 (1998).MathSciNetCrossRefGoogle Scholar - 7.
*Pinus A. G.*, “Positively conditional varieties,” in: Algebra and Model Theory [in Russian], NGTU, Novosibirsk, 2001, 3, pp. 99–106.Google Scholar - 8.
*Pinus A. G.*, “Characterization of conditionally termable functions,” Siberian Math. J.,**38**, No. 1, 136–146 (1997).MathSciNetzbMATHCrossRefGoogle Scholar - 9.
*Pinus A. G.*, “On functions commuting with semigroups of transformations of algebras,” Siberian Math. J.,**41**, No. 6, 1166–1173 (2000).MathSciNetCrossRefGoogle Scholar - 10.
*Pinus A. G.*, “Rational and conditional-rational equivalent algebras,” Algebra and Logic,**41**, No. 3, 182–186 (2002).MathSciNetCrossRefGoogle Scholar - 11.
*Pinus A. G.*, “The positive and generalized discriminators don’t exist,” Discuss. Math., Gen. Algebra Appl.,**20**, 121–128 (2000).MathSciNetzbMATHCrossRefGoogle Scholar - 12.
*Eilenberg S. and Schützenberger M. P.*, “On pseudovarieties,” Adv. Math.,**19**, No. 3, 413–418 (1976).zbMATHCrossRefGoogle Scholar - 13.
*Pinus A. G.*, “Implicitly equivalent universal algebras,” Siberian Math. J.,**53**, No. 5, 862–871 (2012).MathSciNetzbMATHCrossRefGoogle Scholar - 14.
*Pinus A. G. and Zhurkov S. V.*, “On the scales of computability potentials of finite algebras: Results and problems,” J. Math. Sci. (New York),**135**, No. 5, 3363–3376 (2006).MathSciNetCrossRefGoogle Scholar - 15.
*Pinus A. G.*, “The computability potential scale of all finite algebras,” Siberian Math. J.,**48**, No. 3, 539–543 (2007).MathSciNetCrossRefGoogle Scholar - 16.
*Post E. L.*, The Two-Valued Iterative Systems of Mathematical Logic, Princeton University Press, Princeton, N.J. (1941) (Ann. Math. Stud. 5).zbMATHGoogle Scholar - 17.
*Yanov Yu. I. and Muchnik A. A.*, “On the existence of*k*-valued closed classes without a finite basis,” Dokl. Akad. Nauk SSSR,**127**, No. 1, 144–146 (1959).Google Scholar - 18.
*Shevrin L. N. and Ovchinnikova A. Ya.*, Semigroups and Their Subsemigroup Lattices. Parts 1 and 2 [in Russian], Ural Univ., Sverdlovsk (1990–1991).Google Scholar - 19.
*Pinus A. G.*, “Full imbeddings of categories of algebraic systems and determinability of a model by the semigroup of its endomorphisms,” Soviet Math. (Izv. VUZ. Mat.),**26**, No. 1, 97–101 (1982).MathSciNetzbMATHGoogle Scholar - 20.
*Pinus A. G.*, “On the definability of finite algebras by derived categories,” Russian Math. (Izv. VUZ. Mat.),**45**, No. 4, 36–40 (2001).MathSciNetGoogle Scholar - 21.
*Pinus A. G.*, “Definability of locally finite and finite algebras by semigroups of their transformations,” in: Selected Problems of Algebra and Logic [in Russian], AGU, Barnaul, 2007, pp. 173–198.Google Scholar